To whom does the Stone of the Hapless refer?  Why did the Valar preserve this memorial?  Why would they preserve the memorial and not the people themselves?

Túrin and Nienor, I assumed. They're already dead by the time the Valar intervene.

Are Mim’s claims valid?

His claims are valid, perhaps, but why he should want to claim treasure as the last of his people I cannot say. He did betray Túrin.

Why did Hurin kill Mim?  Did Mim deserve to die?

I would say yes, for trying to finish off Beleg. As for betraying the hideaway, that was not the heroic thing to have done, but what choice did he have? Of course, in the situation like that coming, it is only a few of the great heroes who live, and Mîm does not deserve to be counted among that number.

Why was "one thing only" taken by Hurin?

It's an heirloom that is finding its way back to its formers. Húrin is angry that Thingol did not help out.0

”For now my fate is fulfilled, and the purpose of Morgoth achieved; but I am his thrall no longer.”  To what event(s) does now refer?

Probably sowing discord between the Elves and other races. I cannot say for sure.

Was Hurin more despondent upon learning the truth than he was believing Morgoth’s lies?

I cannot answer that, but which is worse: the hopelessness and anger of Morgoth's lies, or seeing how there was hope but that it had been squandered? I cannot say.

What happened to Aure entuluva?

Day did come again. Húrin got a chance to be with his wife one last time. But day is so fleeting, and now it is night again. Day, or at least so bright a day, will never come again.